Three-manifolds with positive Ricci curvature
type: Item
instance of: scholarly article
title: Three-manifolds with positive Ricci curvature
zbMATH Open document ID: 0504.53034
author: Richard S. Hamilton
published in: Journal of Differential Geometry
publication date: 1982-01-01T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.jdg/1214436922
review text: In this paper the following important theorem is proved: Let \(X\) be a compact 3-manifold which admits a Riemannian metric with strictly positive Ricci curvature. Then \(X\) also admits a metric of constant positive curvature. As all manifolds of constant curvature have been completely classified by \textit{J. A. Wolf} in his book [Spaces of constant curvature. New York etc.: McGraw-Hill (1967; Zbl 0162.53304)], by the preceding theorem these are the only compact three-manifolds, which can carry metrics of strictly positive Ricci curvature. Accordingly, a conjecture of \textit{J.-P. Bourguignon} [Global differential geometry and global analysis, Proc. Colloq., Berlin 1979, Lect. Notes Math. 838, 249--250 (1981; Zbl 0437.53011)] is answered affirmatively. As pointed out by the author, this theorem is closely related to the famous Poincaré's conjecture on the compact, simply-connected 3-manifolds and the Smith's conjecture related to the group of covering transformations. If both conjectures were true, the main theorem mentioned above would follow as a corollary. In the proof of this theorem the author studied in advance the equation of evolution \(\frac{\partial g_{ij}}{\partial t}=\frac 2n rg_{ij} - 2R_{ij}\), where \(r\) is the average of the scalar curvature \(R\), namely \(r=\int R\,d\mu/\int\,d\mu\). He proved that if for a compact 3-manifold the initial metric has strictly positive Ricci curvature, then it continues like that for ever, and converges to a metric of constant positive curvature, as \(t\) tends to \(\infty\). Then he made use of the fact peculiar to three dimensions, that the full Riemannian curvature tensor \(R_{hijk}\) can be calculated from the Ricci tensor \(R_{ij}\) (Weyl's conformal curvature tensor is vanishing). In view of this crucial step, the method used here cannot be generalized to the case \(\dim(M)\geq 3\), unless it is extensively modified.
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 3794902
zbMATH Keywords: Ricci-curvature
zbMATH Keywords: equation of evolution
zbMATH Keywords: Nash-Moser inverse function theorem
zbMATH Keywords: method of a priori estimation
zbMATH Keywords: constant positive curvature
Wikidata QID: Q62058975
MaRDI profile type: Publication
OpenAlex ID: W1530161281
cites work: Q3878345
cites work: Q4067705
cites work: Harmonic Mappings of Riemannian Manifolds
cites work: Harmonic maps of manifolds with boundary
cites work: The inverse function theorem of Nash and Moser
cites work: Q5546515
DOI: 10.4310/JDG/1214436922
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description: scientific article; zbMATH DE number 3794902